1. The Unification of Symmetry and Conservation1
The Unification of Symmetry and Conservation
Sergio Pissanetzky

2. The Unification of Symmetry and Conservation – Sergio Pissanetzky2
THE THEORY
I propose a new Theory of Mechanics.
One fundamental principle: Causality.
One postulate: the action functional.
Discrete, scale free.
Describes the system in “high resolution.”
The granularity of description is adjustable.
There are no assumptions of smoothness.
Applies to all dynamical systems.

3. The Unification of Symmetry and Conservation – Sergio Pissanetzky3
PLACING THE THEORY E B F A C G D H
Statistical methods: probabilities.
Differential methods: smoothness.
Causal Mechanics is general.

4. The Unification of Symmetry and Conservation – Sergio Pissanetzky4
MODEL
The model is a causal set.
If not familiar with causal sets think of a computer program.
The action functional is the metric for causal sets.
System specified by variables, states, transitions, trajectories:
variables  elements
transitions  causal relations
trajectories  legal permutations
However, the transition probabilities are irrelevant.

5. The Unification of Symmetry and Conservation – Sergio Pissanetzky5
PRINCIPLE OF SYMMETRY
A causal set always has a symmetry of the action.
The symmetry is represented by the legal permutations.
A causal set always has a conservation law.
A causal set always has a conserved quantity.
Hence, the principle of symmetry follows from causality.
All conserved quantities are determined by the theory.

6. The Unification of Symmetry and Conservation – Sergio Pissanetzky6
PRINCIPLE OF LEAST-ACTION
The action functional is the natural metric of causal sets.
The action depends on the trajectory.
The trajectory is represented by a permutation.
The subset of least-action permutations is a grupoid.
The grupoid has a group-theoretical block system.
The block system is the unique conserved quantity.
Hence, least-action also follows from causality.

7. The Unification of Symmetry and Conservation – Sergio Pissanetzky7
PREDICTIONS
The theory applies to all systems, even the brain.
In 2011, I predicted optimally short dendritic trees in the mammalian brain.
At that time a non-optimal 4/3 power law was accepted.
In 2012, Cuntz proposed a 2/3 optimally short power law.
Hence, the prediction is confirmed.
This is a major success for the theory.
The theory also applies in Physics.
Predicted Noether’s theorem as a particular case of the theory.
Already proved a small part of Noether’s theorem.

8. The Unification of Symmetry and Conservation – Sergio Pissanetzky8
FEATURES
This theory has exceptional features.
It is simple, elegant, and general.
Has only one principle and one postulate.
Does not divide Physics into micro/macro scales.
Does not divide Physics into simple/complex systems.
No limit in the granularity of the description.
Infinitely many conserved quantities, all computable.
Satisfies Smolin rules for a scientific theory.
Is confirmable, falsifiable, and the hypothesis are the simplest among theories.

9. The theory of Causal Mechanics
The Unification of Symmetry and Conservation – Sergio Pissanetzky 9
I call this theory:
The theory of Causal Mechanics